Question: Multiply the following complex numbers, marked as blue dots on the graph: $(2 e^{\pi i / 4}) \cdot (5 e^{5\pi i / 3})$ (Your current answer will be plotted in orange.)
Answer: Multiplying complex numbers in polar forms can be done by multiplying the lengths and adding the angles. The first number ( $2 e^{\pi i / 4}$ ) has angle $\frac{1}{4}\pi$ and radius $2$ The second number ( $5 e^{5\pi i / 3}$ ) has angle $\frac{5}{3}\pi$ and radius $5$ The radius of the result will be $2 \cdot 5$ , which is $10$ The angle of the result is $\frac{1}{4}\pi + \frac{5}{3}\pi = \frac{23}{12}\pi$ The radius of the result is $10$ and the angle of the result is $\frac{23}{12}\pi$.